Ian Sever

In 2014, Princeton mathematician John Conway proposed a simple conjecture:

Let n be a positive integer. Write the prime factorization in the usual way, e.g. 60 = 2² × 3 × 5, in which the primes are written in increasing order, and exponents of 1 are omitted. Then bring exponents down to the line and omit all multiplication signs, obtaining a number f(n). Now repeat.

So, for example, f(60) = f(2² × 3 × 5) = 2235. Next, because 2235 = 3 × 5 × 149, it maps, under f, to 35149, and since 35149 is prime, we stop there forever.

The conjecture, in which I seem to be the only believer, is that every number eventually climbs to a prime. The number 20 has not been verified to do so. Observe that 20 → 225 → 3252 → 223271 → …, eventually getting to more than one hundred digits without reaching a prime!

Conway offered $1,000 for a counterexample, a number that doesn’t “climb to a prime.” The challenge stood for three years before James Davis, who calls himself “not a mathematician by any stretch,” found the pretty 13532385396179 = 13 × 53² × 3853 × 96179, a composite number that loops onto itself and thus never reaches a prime — a find worth $1,000.

I had written about this back in 2006, but it’s worth mentioning again because someone has created this pellucid diagram: Buffalo buffalo Buffalo buffalo buffalo buffalo Buffalo buffalo is a grammatical English sentence. It means something like “Bison residing in Buffalo, New York, feeling themselves intimidated by their fellows, visit a similar fate upon yet others of their local ilk.”

I’d attributed it to linguist William J. Rapaport, but apparently it’s arisen independently at least three times, first (it is believed) by wordplay maven Dmitri Borgmann, in 1965.

I remember, many years ago, when my imagination was warm, and I happened to be in melancholy mood, it distressed me to think of going into a state of being in which Shakespeare’s poetry did not exist. A lady whom I then much admired, a very amiable woman, humored my fancy, and relieved me by saying, ‘The first thing you will meet in the other world will be an elegant copy of Shakespeare’s works presented to you.’ Dr. Johnson smiled benignantly at this, and did not appear to disapprove of the notion.

— James Boswell, Life of Samuel Johnson, 1791

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Sydney Smith to Miss Lucie Austin, July 22, 1835:

Lucy, Lucy, my dear child, don’t tear your frock: tearing frocks is not of itself a proof of genius; but write as your mother writes, act as your mother acts; be frank, loyal, affectionate, simple, honest; and then integrity or laceration of frock is of little import.

And Lucy, dear child, mind your arithmetic. You know, in the first sum of yours I ever saw, there was a mistake. You had carried two (as a cab is licensed to do), and you ought, dear Lucy, to have carried but one. Is this a trifle? What would life be without arithmetic, but a scene of horrors?

You are going to Boulogne, the city of debts, peopled by men who never understood arithmetic; by the time you return, I shall probably have received my first paralytic stroke, and shall have lost all recollection of you; therefore I now give you my parting advice. Don’t marry anybody who has not a tolerable understanding and a thousand a year; and God bless you, dear child.

Sydney Smith

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Here’s something odd: a painting of a young lady riding Aristotle like a pony.

In fact what’s surprising is how thoroughly we’ve forgotten this image, which was once one of the most common artistic motifs of the Northern Renaissance, figuring in scores of paintings, sculptures, and engravings.

The woman is Phyllis, the consort of Alexander the Great, who was a pupil of Aristotle. According to a 13th-century manuscript:

Once upon a time, Aristotle taught Alexander that he should restrain himself from frequently approaching his wife, who was very beautiful, lest he should impede his spirit from seeking the general good. Alexander acquiesced to him. The queen, when she perceived this and was upset, began to draw Aristotle to love her. Many times she crossed paths with him alone, with bare feet and disheveled hair, so that she might entice him.

At last, being enticed, he began to solicit her carnally.

‘This I will certainly not do, unless I see a sign of love, lest you be testing me. Therefore, come to my chamber crawling on hand and foot, in order to carry me like a horse. Then I’ll know that you aren’t deluding me.’

When he had consented to that condition, she secretly told the matter to Alexander, who lying in wait apprehended him carrying the queen. When Alexander wished to kill Aristotle, in order to excuse himself, Aristotle says,

‘If thus it happened to me, an old man most wise, that I was deceived by a woman, you can see that I taught you well, that it could happen to you, a young man.’

Hearing that, the king spared him, and made progress in Aristotle’s teachings.

This is an exemplum, a sort of parable designed to warn the reader away from a bad practice — in this case, allowing passion to overcome reason. But it’s nice to see the lesson taught to Aristotle, who once declared that the capacity for practical reason was undeveloped in children, absent in slaves, and “without authority” in women.

(Thanks, Dan.)

In his 1976 book 100 Numerical Games, French puzzle maven Pierre Berloquin asks whether it’s possible to construct a magic square using the first nine prime numbers (here counting 1 as prime):

1 2 3 5 7 11 13 17 19

Is it?